Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
2.2. A one-period model
and x represent the minimum and the maximum premium, respectively. But the following
reformulation is equivalent and numerically more stable:
g 2 j (xj) = 1 − e −(xj−x) ≥ 0 and g 3 j (xj) = 1 − e −(x−xj) ≥ 0.
The minimum premium x could be justified by a prudent point of view by regulators while
the maximum premium x could be set, e.g., by a consumer right defense association. In the
sequel, we set x = E(Y )/(1 − emin) < x = 3E(Y ), where emin is the minimum expense rate.
Overall, the constraint function gj(xj) ≥ 0 is equivalent to
{xj, gj(xj) ≥ 0} = xj ∈ [x, x], Kj + nj(xj − πj)(1 − ej) ≥ k995σ(Y ) √
nj . (2.5)
2.2.5 Game sequence
For noncooperative games, there are two main solution concepts: Nash equilibrium and
Stackelberg equilibrium. The Nash equilibrium assumes player actions are taken simultaneously
while for the Stackelberg equilibrium actions take place sequentially, see, e.g., Fudenberg
and Tirole (1991); Osborne and Rubinstein (2006). In our setting, we consider the Nash equilibrium
as the most appropriate concept. We give below the definition of a generalized Nash
equilibrium extending the Nash equilibrium with constraint functions.
Definition. For a game with I players, with payoff functions Oj and constraint function gj,
a generalized Nash equilibrium is a vector x ⋆ = (x ⋆ 1 , . . . , x⋆ I ) such that for all j = 1, . . . , I, x⋆ j
solves the subproblem
max
xj
Oj(xj, x ⋆ −j) s.t. gj(xj, x ⋆ −j) ≥ 0.
where xj and x−j denote action of player j and the other players’ action, respectively.
A (generalized) Nash equilibrium is interpreted as a point at which no player can profitably
deviate, given the actions of the other players. When each player’s strategy set does not depend
on the other players’ strategies, a generalized Nash equilibrium reduces to a standard Nash
equilibrium. Our game is a Nash equilibrium problem since our constraint functions gj defined
in Equation (2.4) depend on the price xj only.
The game sequence is given as follows:
(i) Insurers set their premium according to a generalized Nash equilibrium x⋆ , solving for
all j ∈ {1, . . . , I}
x−j ↦→ arg max Oj(xj, x−j).
xj,gj(xj)≥0
(ii) Insureds randomly choose their new insurer according to probabilities pk→j(x ⋆ ): we
get Nj(x).
(iii) For the one-year coverage, claims are random according to a frequency-average severity
model relative to the portfolio size Nj(x ⋆ ).
(iv) Finally the underwriting result is determined by UWj(x ⋆ ) = Nj(x ⋆ )x ⋆ j (1−ej)−Sj(x ⋆ ),
where ej denotes the expense rate.
If the solvency requirement is not fullfilled, in Solvency I, the regulator response is immediate:
depending on the insolvency severity, regulators can withdraw the authorisation to
underwrite new business or even force the company to go run-off or to sell part of its portfolio.
In Solvency II, this happens only when the MCR level is not met. There is a buffer between
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